Multiobjective Programming and Multiattribute Utility Functions in Portfolio Optimization

Multiobjective Programming and Multiattribute Utility Functions in Portfolio Optimization

Matthias Ehrgott, Chris Waters Department of Engineering Science

The University of Auckland

Private Bag 92019, Auckland, New Zealand email: m.ehrgott@auckland.ac.nz Rafail N. Gasimov, Ozden Ustun Department of Industrial Engineering

Osmangazi University Bademlik 26030, Eski¸sehir, Turkey email: {gasimovr,oustun@ogu.edu.tr}

August 14, 2006

Abstract

In recent years portfolio optimization models that consider more criteria than the standard expected return and variance objectives of the Markowitz model have become popular. For such models, two approaches to find a suitable portfolio for an individual investor are possible.

In the multiattribute utility theory (MAUT) approach a utility function is constructed based on the investor’s preferences and an optimization problem is solved to find a portfolio that maximizes the utility function. In the multiobjective programming (MOP) approach a set of efficient portfolios is computed by optimizing a scalarized objective function. The investor then chooses a portfolio from the efficient set. We outline these two approaches using the UTADIS method to construct a utility function and present numerical results for an example.

Keywords: Portfolio optimization; multiobjective programming; multiattribute utility func- tion; UTADIS.

1 Portfolio Optimization

Multicriteria portfolio optimization started with the Markowitz mean-variance model (Markowitz, 1952, 1959). This model assumes that the goal of an average or standard investor is to maximize the (unknown and uncertain) return on investment. The mean-variance model is one possible deterministic substitute of this stochastic optimization problem with the objective to maximize the expected return subject to a constraint on its variance.

Letnbe the number of available assets, let xi andri be the fraction of the available capital invested in and the expected return, respectively of asset i for i= 1, . . . , n. Let and σij be the covariance of the returns of assetsiandj. The optimization problem considered in the Markowitz model is then

maxf1(x) =

n

X

i=1

rixi

subject tof2(x) =

n

X

i=1 n

X

j=1

σijxixj ≤ ε

n

X

i=1

xi = 1

xi ≥ 0 for alli= 1, . . . , n.

(1)

Ifσmin andσmaxare the minimal and maximal attainable values off2(x) it is easy to see that all xthat maximize (1) for some ε ∈ [σmin, σmax] are contained in the set of (weakly) efficient solutions of the biobjective optimization problem

maxf1(x) =

n

X

i=1

rixi

minf2(x) =

n

X

i=1 n

X

j=1

σijxixj n

X

i=1

xi = 1

xi ≥ 0 for alli= 1, . . . , n.

(2)

An efficient solution of (2) is a portfolio, which has the property that when moving to a portfolio with higher return variance will also increase, and when moving to a portfolio with smaller variance, return will decrease, too. Clearly, non-efficient portfolios are undesirable in this context and every efficient portfolio is also an optimal solution to (1) with ε=f2(x). The non-dominated frontier consists of all possible combinations of expected returns and variances of efficient portfolios.

Figure 1 shows the (approximated) non-dominated frontier for an example withn= 40 assets.

This figure is for the data set we will use throughout the paper.

0 50 100 150 200 250 300 350

0 200 400 600 800 1000 1200 1400 1600 f1

f₂

Figure 1: Approximated efficient frontier for an example with n= 40.

While the Markowitz model permeates the field of finance until today, there has been an increasing number of publications that suggest that it is not always appropriate, at least in the case of individual rather than standard investors. We mention a few such publications, but refer to Steuer and Na (2003) for a more detailed analysis of the literature.

Arthur and Ghandforoush (1987) suggest that there are objective and subjective measures for portfolios. Konno (1990) observes that most investors do not actually buy efficient portfolios, but rather those behind the nondominated frontier. Ballestero and Romero (1996) argue for a need to modify the model for average investors in order to approximate the optimal portfolio of an individual investor. Hallerbach and Spronk (1997) explain that most models do not incor- porate the multidimensional nature of the problem and outline a framework for such a view on portfolio management. Finally, Steuer et al. (2006) introduce the “suitable portfolio” investor, who may include objectives other than expected return and variance in their portfolio selection problem. They also explain the fact that investors do buy non-efficient portfolios as the effect of projecting the multidimensional space of the individual investor’s portfolio selection problem to the two-dimensional mean-variance space: The selected portfolios are actually efficient in the higher dimensional space.

In this paper, we have chosen one exemplary multiobjective model of portfolio optimization to illustrate the two approaches to portfolio selection. This is the model presented in Ehrgott et al.(2004), which we can consider as a particular investor’s portfolio selection problem. The five criteria of this model include 12-month performance, 3-year performance, and annual dividend as measures of return. The fourth objective is the Standard and Poor’s star ranking, which describes to what extent an investment fund follows a specific market index and is applied particularly in the case that a portfolio consists exclusively of investment funds, which is the case for the data set we use for numerical experiments. The fifth attribute, the 12-month variance, is used as a measure of the risk of a portfolio.

For asset i ∈ {1, . . . , n} let r_i¹² be the 12-month performance (expected return), let r_i³⁶ be the 36-month (long term) performance, let di ≥ 0 be the relative annual dividend, and let si ∈ {1,2,3,4,5} be the number of stars assigned to asset i (one star indicates relatively poor performance of assets and five stars indicate very good performance). Furthermore letσij be the covariance between the returns of assetsiandj.

We define the following five objective functions.

• f1(x) =Pn

i=1r¹²_i xi is the 12-month performance. This objective function is a measure for the short term expected return.

• f2(x) =Pn

i=1r_i³⁶xi, the 3-year performance, is a measure for the long term expected return.

• f3(x) =Pn

i=1dixi represents the relative annual dividend of a portfolio.

• f4(x) = Pn

i=1sixi is the average star ranking of portfolio x. Standard and Poor’s Fund Service GmbH evaluates the performance of most investment funds contained in their data base on an annual basis which results in a performance ranking.

• f5(x) =−Pn i=1

Pn

j=1σijxixj is the usual variance measure of portfolio risk (we take the negative value so as to maximize all objectives).

Thus, our multiobjective portfolio optimization problem can be written as follows:

max(f1(x), . . . , f4(x), f5(x)) subject to

n

X

i=1

xi = 1

xi ≥ 0 for all i= 1, . . . , n.

(3)

In addition to replacing the two objectives of the Markowitz model (2) by five, we include additional constraints on the minimal and maximal fraction of the capital that can be invested in a single asset and on the number of assets in the portfolio (see also Changet al.(2000)). For that purpose letyi, i= 1, . . . , n denote binary variables with yi = 1 if and only if assetiis contained in the portfolio. With these additions the problem we consider in this paper is

max(f1(x), . . . , f4(x), f5(x)) subject to

n

X

i=1

xi = 1

n

X

i=1

yi = k

xi ≤ riyi for alli= 1, . . . , n xi ≥ liyi for alli= 1, . . . , n xi ≥ 0 for alli= 1, . . . , n

yi ∈ {0,1} for alli= 1, . . . , n.

(4)

The goal of portfolio selection is of course to find the most suitable portfolio for our individual investor. To be able to do this, we have to assume the existence of a utility function for the investor: The utility of a portfolio is a (real-valued) function of the portfolio’s score on the five criteria specified above. It is important to note that this utility function is usually not explicitly available.

The most suitable portfolio is then the one with the highest utility. To compute it there are two possible strategies. In the multiobjective programming approach, we find a set of efficient solutions of (4). The multiattribute utility theory approach is to elicit information from the investor that allows the construction of a utility function. This is then used to convert (4) into a single objective optimization problem which is solved directly to obtain the portfolio with maximal utility.

The two strategies are presented in Sections 2 and 3. In Section 4 we present the results of numerical tests on a dataset withn= 40 sets obtained from the Standard and Poor’s database in 1999.

2 Multiobjective Programming

In this section we introduce some definitions and results of multiobjective programming. A mul- tiobjective programming problem can be written as

maxx∈Xf(x), (5)

whereX ⊂Rⁿis the feasible set in decision spaceRⁿ andf :Rⁿ→R^pis a vector valued objective function mapping a feasible solutionxto a pointf(x) = (f1(x), . . . , fp(x)) in objective spaceR^p. We denote byY :=f(X) the feasible set in objective space.

LetR^p₌ :={y∈R^p:yk =0, k= 1, . . . , p}andR^p> :={y ∈R^p:yk >0, k= 1, . . . , p}. For any y¹, y²∈R^p we define

y¹ 5 y²ify²−y¹∈R^p₌, y¹ ≤ y²ify¹5y² andy¹6=y², y¹ < y²ify²−y¹∈intR^p₌=R^p>. In the context of portfolio optimization we will have

X ={(x, y)∈Rⁿ₌× {0,1}ⁿ:e^Tx= 1;liyi≤xi≤riyi fori= 1, . . . , n}.

Definition 1 Let Y be a non-empty subset of R^p.

1. An elementy∈Y is called non-dominated if ({y}+R^p₌)∩Y ={y}, i.e. there is noy⁰∈Y such that y⁰≥y.

2. An element y ∈ Y is called properly non-dominated (in the sense of Benson) if y is a non-dominated element of Y and the zero element of R^p is a non-dominated element of cl cone(Y −R₌^p −y), wherecl(Y)denotes the closure of a set Y andcone(Y) :={αy:α≥ 0, y∈Y}.

The set of all non-dominated elements ofY is denotedYN, the set of all properly non-dominated elementsYpN.

Definition 2 A feasible solution x∈ X is called (properly) efficient if y =f(x) is a (properly) non-dominated element of Y.

The set of (properly) efficient solutions of a multiobjective programming problem is denoted XE (XpE).

Multiobjective programming problems are generally solved through scalarization. The mul- tiobjective programme (5) is transformed into a single objective problem minx∈Xs(x, λ), the objective function of which depends on parameter λ. Solving the single objective problem for a range of parameter values yields (some) efficient solutions to the multiobjective programme. Many scalarization methods are known, see the survey in Ehrgott and Wiecek (2005). The main issues for solving a multiobjective programme by scalarization are to show that a) every optimal solu- tion to the scalarized problem is efficient and b) for every efficient solution ˆxto a multiobjective programme there is a parameter ˆλsuch that ˆxis an optimal solution to minx∈Xs(x,λ).ˆ

In this paper we use the scalarization of Gasimov (2001). He introduces a class of increasing convex functions to scalarize the multiobjective programme (5) without any assumptions on ob- jectives and constraints of the problem under consideration. Another advantage of this approach is that it preserves convexity, if the objective functions of the initial problem are linear or convex.

Now we briefly present the main scalarization results of Gasimov (2001). Let W :=n

(α, w)∈R×R^p₌: 0≤α <min{w1, . . . , wp}o

. (6)

Theorem 1 (Gasimov (2001)) Suppose that for some (α, w)∈W a feasible solution xˆ∈X is an optimal solution to the scalar maximization problem

maxx∈X α

p

X

k=1

|fk(x)|+

p

X

k=1

wkfk(x)

!

. (7)

thenxˆ is a Benson properly efficient solution to (5).

Theorem 2 (Gasimov (2001)) Let xˆ ∈X be a Benson proper efficient solution to (5). Then there exists a vector (α, w) ∈ W such that ˆx is an optimal solution to the scalar maximization problem

maxx∈X α

p

X

k=1

|fk(x)−fk(ˆx)|+

1

X

k=1

wk(fk(x)−fk(ˆx)) (8) In non-convex multiobjective programmes the distinction between supported and non-support- ed efficient solutions is important. An efficient solution ˆx ∈XE is called supported, if there is w∈R^p> such that ˆxis an optimal solution to

maxx∈X p

X

k=1

wkfk(x).

It is well known (Geoffrion, 1968) that ifX is convex and allfk, k= 1, . . . , pare convex functions, then all Benson proper efficient solutions are supported, see e.g. Ehrgott (2005). However, for non-convex problems there exist non-supported efficient solutions. This observation yields the following Lemma.

Lemma 1 Let Y ⊂ R^p be a non-empty set, let yˆ ∈ Y be a non-dominated point of Y, and let y¹, . . . , y^m∈Y. If there is someβ∈R^p> with Pp

k=1βk= 1 such thaty^β=Pp

k=1βkyk <yˆthenyˆ is not an optimal solution to the weighted sum problemmaxy∈Y λ^Ty, whereλ∈R^p>.

It is evident that if ˆx∈ X is an efficient solution to problem (5) then it is also an efficient solution to the shifted multiobjective programme

maxx∈X(f1(x)−a1, . . . , fp(x)−ap), (9) where a ∈R^p is an arbitrary vector. Such a shifting can be used in situations when objectives do not change sign on the whole feasible set X in order to make the absolute value used in the scalarized problem (7) sensible. In this case we can formulate the following scalarized problem, which is similar to that in (8), and can be used even if we do not know any efficient solution.

maxx∈X α

p

X

k=1

|fk(x)−ak|+

p

X

k=1

wk(fk(x)−ak)

!

. (10)

We can therefore completely characterize Benson proper efficient solutions through Gasimov’s scalarization.

Corollary 1 A feasible solutionxˆ∈X is Benson proper efficient if and only if there area∈R^p, α∈R, andw∈R^p₌ with (α, w)∈W such that xˆ is an optimal solution to

maxx∈X α

p

X

k=1

|fk(x)−ak|+

p

X

k=1

wk(fk(x)−ak)

!

. (11)

Note that for α= 0 (11) reduces to the weighted sum scalarization maxx∈XPp

k=1wkfk(x).

Therefore non-supported efficient solutions can only be found withα >0.

3 Multiattribute Utility Theory and the UTADIS Method

In the multiattribute utility theory (MAUT) approach to portfolio optimization the goal is to construct a utility function, that assigns any portfoliox∈X a utility value. The utility function is a function of the scores of a portfolio on the selected criteria or attributes. The portfolio optimization problem is then solved by finding a portfolio that maximizes the utility function.

According to Keeney and Raiffa (1993) the set of attributes should becomplete, operational, decomposable, non-redundant, and minimal. It is well known that an additive utility function exists if the attributes are mutually preferentially independent Keeney and Raiffa (1993). This means that the conditional preferences of one attribute given a second attribute do not depend on the value of the second attribute. We shall assume that an additive utility function exists.

Thus, we assume that there is a functionU :R^p→Rthat maps an outcome vectory∈R^p to a global utility valueU(y)∈R. U has the form

U(y) =

p

X

k=1

µku˜k(yk), (12)

where the marginal utility functions ˜uk satisfy 0≤u˜k(yk)≤1 for ally∈Y andk= 1, . . . , pand µk is an importance weight of marginal utility function ˜uk.

In this section we explain how to construct U using the UTADIS method as described in Doumpos and Zopounidis (2002). The UTADIS method (Utilit´es Additives Discriminates) is a method developed for the classification of a finite set of alternatives x^j with attribute vectors y^j into q predefined ordered classes Cl, where C1 and Cq contain the most and least preferred alternatives, respectively. This is done by constructing an additive utility functionU as in (12) and

utility thresholds ¯ul∈Rforl= 1, . . . , q−1 such thatx^jis assigned to classClif ¯ul≤U(y^j)<u¯l−1

(here ¯u0=∞and ¯uq =−∞).

A reference set{x^j : j = 1, . . . , o} of alternatives is selected and classified into the q classes according to the values y^j_k = fk(x^j) by the decision maker. We denote by ml the number of alternatives of the reference set in classCl. An optimization model is then formulated to determine the marginal utility functions ˜uk, their weights µk and the utility thresholds ¯ul. The objective of this optimization problem is the minimization of the classification error rate on the reference set. If the marginal utilities ˜uk are approximated by piecewise linear functions, the optimization problem turns into a linear programme.

For ease of exposition we assume now that all ˜uk are increasing, i.e. more is preferred to less for all attributesyk, and let [yk∗, y^∗_k] be the possible range of values. To further simplify notation, we let uk = µku˜k (so that µk = uk(y_k^∗) and uk(yk∗) = 0). For each of the p criteria yk, let y_k^h, h = 1, . . . , hk denote the breakpoints of the piecewise linear function uk with y¹_k = yk∗ and y_k^hk=y^∗_k. Then

uk(yk) =

h−1

X

r=1

wkr+ yk−y_k^h y_k^h+1−y_k^hwkh⁰_k

for all yk ∈[y_k^h, y^h+1_k ] and appropriately chosen constantswkr =uk(y_k^h+1)−uk(y^h). The global utility of an alternativexwith criteria or attribute vectory=f(x) is then

U(y) =

p

X

k=1





h⁰_k−1

X

r=1

wkr+ yk−y_k^h0k y_k^h0k+1−y^h_k^0k

wkh⁰_k



,

whereh⁰_k denotes the interval [y^h_k^0k, y_k^h0k+1] in which the valueyk falls.

Lettingσ⁺_j := max{0,u¯l−U(y^j)} andσ⁻_j := max{0, U(y^j)−u¯l−1}fory ∈Cl, l= 1, . . . , qbe the classification errors for alternativex^j in the reference set, we can formulate the LP

min

q

X

l=1

1 ml



 X

x^j∈C_l

(σ⁺_j +σ⁻_j)



 (13)

p

X

k=1





h⁰_k−1

X

r=1

wkr+ y^j_k−y^h_k^0k y_k^h0k+1−y_k^h0k

wkh⁰_k



−u¯1+σ_j⁺ ≥ δ1 x^j ∈C1 (14)

p

X

k=1





h⁰_k−1

X

r=1

wkr+ y^j_k−y^h_k^0k y^h_k^0k+1−y_k^h0k

wkh⁰_k



−¯ul+σ_j⁺ ≥ δ1 x^j ∈Cl,2≤l≤q−1 (15)

p

X

k=1





h⁰_k−1

X

r=1

wkr+ y_k^j−y_k^h0k y_k^h0k+1−y^h_k^0k

wkh⁰_k



−u¯l−1−σ_j⁻ ≤ −δ2 x^j∈Cl,2≤l≤q−1 (16)

p

X

k=1





h⁰_k−1

X

r=1

wkr+ y_k^j−y_k^h0k y_k^h0k+1−y^h_k^0k

wkh⁰_k



−u¯q−1−σ_j⁻ ≤ −δ2 x^j∈Cq (17)

p

X

k=1 hk−1

X

h=1

wkh = 1 (18)

¯

ul−u¯l+1 ≥ s l= 1, . . . , q−2 (19) σ⁺_j, σ_j⁻ ≥ 0 j= 1, . . . , o (20)

wkh ≥ 0 k= 1, . . . , p

h= 1, . . . , hk−1 (21)

Constraints (14) – (17) define the classification errorsσ_j⁺ andσ⁻_j. (18) normalizes the utility functionU so thatU(y^∗) = 1, (19) makes sure that utility thresholds are different. The parameters δ1and δ2 are small positive numbers to avoid cases whereU(y^j) = ¯uk forx^j ∈Cl. sis chosen to be bigger thanδ1andδ2and guarantees that ¯ul+1 is greater than ¯ul. The variables in the LP are wkr, ¯ul, andσ⁺_j, σ_j⁻.

The utility functionU(y) was used to formulate a mixed integer non-linear programme maxU(f(x)) =

5

X

k=1

µku˜k(fk(x))

subject to

n

X

i=1

xi = 1

n

X

i=1

yi = k

xi ≤ riyi for alli= 1, . . . , n xi ≥ liyi for alli= 1, . . . , n xi ≥ 0 for alli= 1, . . . , n yi ∈ {0,1} for alli= 1, . . . , n.

(22)

We address the question, whether an optimal solution of (22) is an efficient solution to (4).

Proposition 1 Let U : R^p → R be componentwise increasing and assume that U attains its maximum aty^∗. Theny^∗ is a non-dominated element of Y.

Proof: LetU(y^∗) = max{U(y) :y∈Y}. Ify^∗is dominated there exists somey∈Y such that yk ≥ y_k^∗ for allk = 1, . . . , p and yj > y^∗_j for at least one index j. BecauseU is componentwise increasing, we haveU(y)> U(y^∗), a contradiction that implies thaty^∗ is nondominated. 2 Note that the functionU used in (22) is componentwise increasing because the marginal utility functions ˜uk are all increasing. Then the fact thatYN =f(XE) implies that an optimal solution of (22) is an efficient solution of (4).

We also want to study the effect of the cardinality of the portfolio on its utility. That can be done by solving (22) for all feasible values of k. In order to obtain the portfolio with maximal utility, independent of the cardinality, but with the lower and upper bounds onxi, we remove the constraintPn

i=1yi=k.

4 Results

We have used a dataset ofn= 40 investment funds from Standard and Poor’s 1999 database. For the numerical tests we have considered an unconstrained problem (i.e. the number of assets in the portfolio was not prescribed and all lower boundsli= 0, all upper boundsri= 1) as well as a constrained problem with the valuesli= 0.05, ri= 0.3 to limit the fraction of an asset in a portfolio andk= 10 for cardinality constrained portfolios. For both problems we have solved the scalarized MOP (11) and the non-linear (mixed integer) problem (22). All problems were solved using the GAMS/MINOS derivative-free nonlinear programming (DNLP) solver and GAMS/DICOPT mixed integer nonlinear programming (MINLP) solver. Documentation and information about GAMS and its solvers are available on the Internet atwww.gams.com. All programs have been run on a HP Workstation xw6000 with WINDOWS operating system on two processors.

For the unconstrained portfolio problem we have useda= (234.46,175.3,4.51,3,0) and various values of αand w in (11) to find efficient solutions to problem (3) corresponding to some non- dominated points. The non-dominated points are presented in Table 1.

Table 1: Non-dominated points of the unconstrained problem found by solving (10).

No. α w1 w2 w3 w4 w5 f1 f2 f3 f4 f5

1 0 1 0 0 0 0 432.89 333.20 0.00 4.00 1.00

2 0 0 1 0 0 0 234.46 336.49 0.00 4.00 1.00

3 0 0 0 1 0 0 11.40 37.64 8.55 1.00 1.00

4 0 0 0 0 1 0 15.96 57.69 2.93 5.00 0.83

5 0 0 0 0 0 1 97.19 124.39 1.87 2.83 0.39

6 0 1 1 1 1 1 432.89 333.20 0.00 4.00 1.00

7 0 1 1 1 1 5 432.89 333.20 0.00 4.00 1.00

8 0 1 1 1 1 10 432.89 333.20 0.00 4.00 1.00

9 0 1 1 1 1 20 432.89 333.20 0.00 4.00 1.00

10 0 1 1 1 1 50 432.89 333.20 0.00 4.00 1.00

11 0 1 1 1 1 1000 275.68 261.98 0.11 4.07 0.58

12 0 4 6 30 30 30 432.89 333.20 0.00 4.00 1.00

13 0 5 5 30 5 5 432.89 333.20 0.00 4.00 1.00

14 0 1 1 2 1000 2000 82.73 119.54 0.47 5.00 0.54

15 0 1 1 5 2000 6000 83.17 116.76 0.45 4.98 0.53

16 0 1 1 1000 1000 1000 0.90 24.55 5.34 5.00 1.00

17 0 1 1 250 750 1000 20.16 88.79 4.73 5.00 0.77

18 0 250 250 5000 250 5000 432.89 333.20 0.00 4.00 1.00

19 0 1000 5000 250 1000 1000 432.89 333.20 0.00 4.00 1.00

20 0 1 2 1 2 1 432.89 333.20 0.00 4.00 1.00

21 4 4 5 6 7 8 308.02 335.27 0.00 4.00 0.78

22 5 9 8 7 6 5 432.89 333.20 0.00 4.00 1.00

23 1 1 1 1 1 1 234.48 193.91 3.93 2.70 0.76

24 4 12 11 10 9 8 432.89 333.20 0.00 4.00 1.00

25 1 8 8 8 8 2 432.89 333.20 0.00 4.00 1.00

26 4 10 5 10 5 10 432.89 333.20 0.00 4.00 1.00

27 3 3 4 5 3 4 287.52 335.61 0.00 4.00 0.82

28 1 5 1 2 3 4 432.89 333.20 0.00 4.00 1.00

29 2 5 2 2 3 4 432.89 333.20 0.00 4.00 1.00

30 3 6 9 9 9 9 432.89 333.20 0.00 4.00 1.00

31 1 20 50 20 50 20 432.89 333.20 0.00 4.00 1.00

32 10 21 51 21 51 21 432.89 333.20 0.00 4.00 1.00

33 1 1 10 10 1 50 274.21 335.83 0.00 4.00 0.86

34 2 20 5 4 3 2 432.89 333.20 0.00 4.00 1.00

35 0 2 10 10 10 100 432.89 333.20 0.00 4.00 1.00

36 5 5 15 15 5 5 234.46 336.49 0.00 4.00 1.00

37 4 4 8 16 32 64 315.78 335.14 0.00 4.00 0.77

38 4 4 6 30 30 30 315.78 335.14 0.00 4.00 0.77

39 5 5 5 30 30 5 234.47 194.08 4.02 2.59 0.76

40 1 1 1 2 1000 2000 114.36 156.87 0.56 4.89 0.57

41 1 1 1 5 2000 6000 90.08 128.44 0.43 4.94 0.52

42 1 1000 1 1 1 1 432.89 333.20 0.00 4.00 1.00

43 1 1 1000 1 1 1 234.46 336.49 0.00 4.00 1.00

44 1 1 1 1000 1 1 11.40 37.64 8.55 1.00 1.00

45 1 1 1 1 1000 1 153.12 175.77 0.00 5.00 1.00

46 1 1 1 1 1 1000 234.46 218.40 0.18 3.85 0.51

47 0.25 16.27 0.25 3.21 53.52 26.75 432.89 333.20 0.00 4.00 1.00

In Table 2 eleven efficient solutions obtained for nonzero values of the parameterαextracted from Table 1 are presented. As it can be seen from Table 2 different efficient solutions have been obtained for the cases α = 0 and α 6= 0. It is remarkable that the conic scalarization method provides different solutions for the same set of preference weights, while very similar solutions have been calculated for different weights withα= 0.

Table 2: Non-dominated points of the unconstrained problem found by solving (10) with zero and nonzeroα.

No. α w1 w2 w3 w4 w5 f1 f2 f3 f4 f5

21 4 4 5 6 7 8 308.02 335.27 0.00 4.00 0.78

0 4 5 6 7 8 432.89 333.20 0.00 4.00 1.00

23 1 1 1 1 1 1 234.48 193.91 3.93 2.70 0.76

0 1 1 1 1 1 432.89 333.20 0.00 4.00 1.00

27 3 3 4 5 3 4 287.52 335.61 0.00 4.00 0.82

0 3 4 5 3 4 432.89 333.20 0.00 4.00 1.00

33 1 1 10 10 1 50 274.21 335.83 0.00 4.00 0.86

0 1 10 10 1 50 432.89 333.20 0.00 4.00 1.00

37 4 4 8 16 32 64 315.78 335.14 0.00 4.00 0.77

0 4 8 16 32 64 432.89 333.20 0.00 4.00 1.00

38 4 4 6 30 30 30 315.78 335.14 0.00 4.00 0.77

0 4 6 30 30 30 432.89 333.20 0.00 4.00 1.00

39 5 5 5 30 30 5 234.47 194.08 4.02 2.59 0.76

0 5 5 30 30 5 432.89 333.20 0.00 4.00 1.00

40 1 1 1 2 1000 2000 114.36 156.87 0.56 4.89 0.57

0 1 1 2 1000 2000 82.73 119.54 0.47 5.00 0.54

41 1 1 1 5 2000 6000 90.08 128.44 0.43 4.94 0.52

0 1 1 5 2000 6000 83.17 116.76 0.45 4.98 0.53

45 1 1 1 1 1000 1 153.12 175.77 0.00 5.00 1.00

0 1 1 1 1000 1 152.86 176.94 0.00 5.00 1.00

46 1 1 1 1 1 1000 234.46 218.40 0.18 3.85 0.51

0 1 1 1 1 1000 275.68 261.98 0.11 4.07 0.58

In Table 3 we show non-dominated points obtained for the constrained problem (4).

Table 3: Non-dominated points of the constrained problem found by solving (10).

No. α w1 w2 w3 w4 w5 f1 f2 f3 f4 f5

1 0 1 0 0 0 0 280.02 240.80 0.29 4.15 0.73

2 0 0 1 0 0 0 243.00 278.80 0.35 4.25 0.67

3 0 0 0 1 0 0 17.64 43.09 5.94 2.40 0.79

4 0 0 0 0 1 0 45.31 133.97 2.11 5.00 0.81

5 0 0 0 0 0 1 97.98 127.38 1.41 3.11 0.39

6 0 1 1 1 1 1 272.96 258.01 0.35 4.30 0.71

7 0 1 1 1 1 5 272.96 258.01 0.35 4.30 0.71

8 0 1 1 1 1 10 272.96 258.01 0.35 4.30 0.71

9 0 1 1 1 1 20 270.77 259.93 0.35 4.30 0.69

10 0 1 1 1 1 50 259.04 270.27 0.35 4.30 0.64

11 0 1 1 1 1 1000 233.22 240.88 0.20 4.02 0.53

12 0 4 6 30 30 30 252.73 275.84 0.35 4.30 0.63

13 0 5 5 30 5 5 272.96 258.01 0.35 4.30 0.71

14 0 1 1 2 1000 2000 82.33 120.03 0.81 5.00 0.54

15 0 1 1 5 2000 6000 86.54 127.54 0.56 4.95 0.53

16 0 1 1 1000 1000 1000 22.06 65.35 5.47 3.50 0.76

17 0 1 1 250 750 1000 53.13 114.57 3.04 5.00 0.64

18 0 250 250 5000 250 5000 270.67 256.81 0.58 4.20 0.70

19 0 1000 5000 250 1000 1000 247.30 278.15 0.37 4.25 0.64

20 0 1 2 1 2 1 252.73 275.84 0.35 4.30 0.63

21 4 4 5 6 7 8 241.78 278.52 0.60 4.15 0.67

22 5 9 8 7 6 5 272.96 258.01 0.35 4.30 0.71

23 1 1 1 1 1 1 234.46 210.38 2.29 3.52 0.65

24 4 12 11 10 9 8 272.96 258.01 0.35 4.30 0.71

25 1 8 8 8 8 2 272.96 258.01 0.35 4.30 0.71

26 4 10 5 10 5 10 278.96 247.32 0.37 4.20 0.72

27 3 3 4 5 3 4 241.78 278.52 0.60 4.15 0.67

28 1 5 1 2 3 4 280.02 240.80 0.29 4.15 0.73

29 2 5 2 2 3 4 280.02 240.80 0.29 4.15 0.73

30 3 6 9 9 9 9 252.73 275.84 0.35 4.30 0.63

31 1 20 50 20 50 20 247.30 278.15 0.37 4.25 0.64

32 10 21 51 21 51 21 247.30 278.15 0.37 4.25 0.64

33 1 1 10 10 1 50 241.78 278.52 0.60 4.15 0.67

34 2 20 5 4 3 2 278.96 247.32 0.37 4.20 0.72

35 0 2 10 10 10 100 247.3 278.15 0.37 4.25 0.64

36 5 5 15 15 5 5 241.78 278.52 0.60 4.15 0.67

37 4 4 8 16 32 64 241.78 278.52 0.60 4.15 0.67

38 4 4 6 30 30 30 241.09 277.79 0.64 4.20 0.67

39 5 5 5 30 30 5 234.46 217.31 2.19 3.70 0.67

40 1 1 1 2 1000 2000 148.25 175.3 0.75 4.78 0.52

41 1 1 1 5 2000 6000 88.84 130.92 0.56 4.94 0.52

42 1 1000 1 1 1 1 280.02 240.80 0.29 4.15 0.73

43 1 1 1000 1 1 1 243.00 278.80 0.35 4.25 0.67

44 1 1 1 1000 1 1 25.93 73.81 5.92 2.40 0.68

45 1 1 1 1 1000 1 139.46 160.22 0.55 4.95 0.72

46 1 1 1 1 1 1000 234.46 218.58 0.54 3.80 0.53

47 0.25 16.27 0.25 3.21 53.52 26.75 280.02 240.80 0.29 4.15 0.73

For the multiattribute utility model (22) we have to discuss the assumptions first. We have to mention that our five attributes do probably not entirely satisfy the decomposability and non- redundancy (12-month and 3-year performance are certainly correlated to some extent and the Standard & Poor’s star ranking measures both risk and return), this was accepted for this study.

We are also aware of the fact that due to correlation among some of the attributes mutual prefer- ential independence is probably not completely satisfied. Nevertheless, we assume that an additive utility function exists for this problem.

We have used the set of single asset portfolios as reference set, i.e. x^j = e^j, wheree^j is the j^th unit vector in Rⁿ and three classes, i.e. q = 3. The classification of the reference set was done by one of the authors (Waters). The breakpoints y_k^h were chosen such that, after ranking the alternatives according to criterion k, an equal number of alternatives would fall into each interval [y^h_k, y_k^h+1]. We chosehk = 6, i.e. 5 intervals for each criterion and chose the parameters δ1 =δ2 = 0.0001 ands = 2δ1. After solving LP (13) – (21) we have also performed sensitivity analysis as recommended in Doumpos and Zopounidis (2002). This involved solving additional LPs with the objective to minimize the utility thresholds ¯ul and the weightsµk of the objectives, respectively, with a constraint that the classification error is at most 5% worse than in the original LP (13) – (21). The solutions of all LPs were very similar.

We have averaged the results to obtain the final utility function U. The details are given in Table 4.

Table 4: Breakpointsy^h_k and weightsµk of utility functions ˜uk.

12-month 3-year Dividend Variance Star Ranking

yk u˜k yk u˜k yk u˜k yk u˜k yk u˜k

0 0.000 0 0.000 0.0 0.000 0 100.000 0 0.000

15 3.966 40 97.558 0.5 1.651 5 96.632 2 98.671

40 3.966 90 100.000 1.5 16.350 15 96.584 3 98.671

60 3.966 130 100.000 2.5 16.835 20 95.672 4 99.706

150 3.966 180 100.000 4.5 16.835 30 90.207 5 100.000

450 100.000 350 100.000 10.0 100.000 50 0.000

µk 16.27 0.25 3.21 26.75 53.52

The piecewise linear utility functionsuk were implemented as follows. Let y_k¹, . . . , y^h_k^k be the breakpoints. Thenyk =y^h_kzh+y^h+1_k (1−zh) for some 0≤zh≤1 andy_k^h≤yk≤y_k^h+1. Thus

uk(yk) =uk(yk^h)zh+uk(y_k^h+1)(1−zh) and the piecewise linear functionuk is modeled by adding the constraints

zh ≤ wh−1+wh h= 1, . . . , hk h_k

X

h=1

wh = 1

hk

X

h=1

zh = 1

h_k

X

h=1

y^h_kzh = yk

wh ∈ {0,1} h= 1, . . . , hk

zh ≥ 0 h= 1, . . . , hk

withw0=whk= 0.

Table 5: Points of the constrained problem found by solving (22) with different weights.

No. µ1 µ2 µ3 µ4 µ5 U(y) f1 f2 f3 f4 f5

1 0.163 0.003 0.032 0.535 0.268 87.80 280.02 240.80 0.29 4.15 0.73

2 1 0 0 0 0 45.59 280.02 240.80 0.29 4.15 0.73

3 0 1 0 0 0 100.00 280.02 240.80 0.29 4.15 0.73

4 0 0 1 0 0 38.61 17.64 43.09 5.94 2.40 0.79

5 0 0 0 1 0 99.74 97.98 127.38 1.41 3.11 0.39

6 0 0 0 0 1 100.00 58.48 90.00 1.50 5.00 0.60

7 1 1 1 1 1 70.92 260.21 221.98 1.50 3.74 0.65

8 2 2 1 1 1 70.79 269.00 230.49 1.20 3.75 0.67

9 2 1 2 1 1 58.66 260.38 221.67 1.50 3.64 0.66

10 2 1 1 1 2 70.73 269.00 230.49 1.20 3.75 0.67

11 2 1 1 2 1 70.74 269.29 237.69 1.17 3.95 0.68

12 1 2 2 1 1 68.22 27.83 77.96 5.90 2.55 0.68

13 1 2 1 1 2 79.24 260.21 221.98 1.50 3.74 0.65

14 1 2 1 2 1 79.25 259.45 228.40 1.50 3.93 0.66

15 1 1 2 1 2 68.27 25.93 73.81 5.92 2.40 0.68

16 1 1 2 2 1 68.14 25.93 73.81 5.92 2.40 0.68

17 1 1 1 2 2 79.18 259.45 228.40 1.50 3.93 0.66

For different weights in (22) we obtained the results shown in Table 5.

The optimal solution of the constrained problem is x1 = x2 = 0.3, xi = 0.05, i = 3, . . . ,10 withf(x) = (280.02,240.8,0.29,4.5,0.73) with utilityU(y) = 87.8. It is remarkable that the same result was obtained by using the scalarization and utility function approaches. See the first row of Table 5 and the last row of Table 3, for example.

The optimal solution of the unconstrained problem isx1= 1 withf(x) = (432.89,333.20,0,4,1) with utilityU(y) = 95.67.Note that this solution has also been calculated using the scalarization approach with the weights (4,5,6,7,8) or (1,1,1,1,1), for example (see Table 2).

It is worth mentioning that using the conic scalarization method new and different efficient solutions have been obtained for the same set of weights (for different values of α). This means that the conic scalarization method provides several solutions for the given preferences and the resulting solution representing the decision maker’s preferences can be chosen among these.

The optimal solution of the problem with optimal portfolio size isx1=x2=x3= 0.3, x4= 0.1, i.e. k= 4 withf(x) = (311.69,291,0,4.1,0.72) and utilityU(y) = 89.41.

Solving (22) for all feasible values ofkwe obtained the results shown in Table 6. Note that (22) is infeasible fork≤3 since ri = 0.3. Figure 2 shows the decreasing utility values with increasing portfolio cardinality.

Table 6: Optimal utility values for all feasible cardinalities.

k U(y) f1 f2 f3 f4 f5

4 89.41 311.69 291.00 0.00 4.10 0.72

5 89.37 310.90 294.57 0.00 4.10 0.71

6 89.15 306.39 284.31 0.03 4.15 0.69

7 88.86 300.65 271.57 0.03 4.20 0.70

8 88.54 294.13 257.84 0.28 4.15 0.71

9 88.17 287.28 244.31 0.28 4.15 0.73

10 87.80 280.02 240.80 0.29 4.15 0.73

11 87.19 268.06 237.35 0.37 4.20 0.72

12 86.54 254.59 232.86 0.56 4.25 0.71

13 85.93 239.39 221.49 0.99 4.10 0.67

14 85.32 226.68 214.05 1.10 4.10 0.67

15 84.70 212.61 207.73 1.37 4.05 0.66

16 83.75 193.85 200.17 1.40 4.10 0.65

17 82.79 175.21 191.10 1.42 4.10 0.64

18 81.83 156.49 181.65 1.43 4.15 0.64

19 81.63 66.33 115.80 1.50 4.80 0.62

≥20 81.61 67.58 121.21 1.51 4.70 0.62

Maximal utility values for different portfolio size

76 78 80 82 84 86 88 90

4 8 12 16 20 24 28 32 36 40

Portfolio size (k) U(y)

Figure 2: The maximal utility of a portfolio decreases with increasing cardinality.

5 Conclusion

In this paper we have explained two approaches to portfolio optimization. They differ in the sequence of elicitation of investor preference and optimization. While the multiobjective program- ming approach uses optimization first to find efficient solutions of a portfolio selection problem with multiple criteria for the investor to choose from, the multiattribute utility approach elicits preference information from the investor to construct a utility function which is subsequently opti- mized. While, under reasonable assumptions, both approaches will yield portfolios with the same utility, they do have quite unique challenges.

The main challenge for the multiobjective programming approach is of a computational nature:

Can (a representative subset of) the efficient solutions be computed, so that selecting a portfolio from this set guarantees a most preferred portfolio for the investor? If more than two criteria are used, computing all efficient solutions of a non-linear mixed integer programme such as (4) is not currently possible. There is some research on computing a set of representative efficient solutions, but this is restricted to linear programmes or integer linear programmes with two objectives. Such a set should reflect all the possible trade-offs between the criteria available in the set of efficient solutions, but be small enough to allow inspection by the investor. The advantage is clearly that the only assumption on the investor’s utility function is that the criteria represent all relevant attributes of a portfolio.

For the multiattribute utility approach the major questions are of a methodological nature: Is an additive utility function justified? IsUsensitive to the choice of the reference set? What method for construction of the utility function and preference elicitation should be used? For example, for linear criteria functions fk it is clear that there are single assets for which the minimal and maximal values offk(x) are attained, but for the variance this is not the case. Hence choosing the set of assets as reference set might give a false range of values in the construction ofuk. The major advantage of the approach is that only one optimization is necessary to find the most preferred portfolio.

In conclusion, the approach to solution of a portfolio optimization problem must be carefully considered by the investor.

References

Arthur, L. and Ghandforoush, P. (1987). Subjectivity and portfolio optimization. In K. Lawrence, J. Guerard, and G. Reeves, editors, Advances in Mathematical Programming and Financial Planning, pages 171–186. JAI Press, Greenwich, CO.

Ballestero, E. and Romero, C. (1996). Portfolio selection: A compromise programming solution.

Journal of the Operational Research Society,47, 1377–1386.

Chang, T., Meade, N., Beasley, J., and Sharaiha, Y. (2000). Heuristics for cardinality constrained portfolio optimisation. Computers and Operations Research,27, 1271–1302.

Doumpos, M. and Zopounidis, C. (2002). Multicriteria Decision Aid Classification Methods, vol- ume 73 ofApplied Optimization. Kluwer Academic Publishers, Dordrecht.

Ehrgott, M. (2005). Multicriteria Optimization. Springer-Verlag, Berlin, 2nd edition.

Ehrgott, M. and Wiecek, M. (2005). Multiobjective programming. In J. Figueira, S. Greco, and M. Ehrgott, editors,Multicriteria Decision Analysis: State of the Art Surveys, pages 667–722.

Kluwer Academic Publishers, Boston.

Ehrgott, M., Klamroth, K., and Schwehm, C. (2004). An MCDM approach to portfolio optimiza- tion. European Journal of Operational Research, 155, 752–770.

Gasimov, R. (2001). Characterization of the benson proper efficiency and scalarization in noncon- vex vector optimization. In M. Koksalan, S. Zionts, and R. Steuer, editors,Multiple Criteria

Decision Making in the New Millenium, volume 507 ofLecture Notes in Economics and Math- ematical Systems, pages 189–198.

Geoffrion, A. (1968). Proper efficiency and the theory of vector maximization. Journal of Mathe- matical Analysis and Applications, 22, 618–630.

Hallerbach, W. and Spronk, J. (1997). A multi-dimensional framework for portfolio management.

In M. H. Karwan, J. Spronk, and J. Wallenius, editors,Essays in Decision Making. A Volume in Honour of Stanley Zionts, pages 275–293. Springer, Berlin.

Keeney, R. L. and Raiffa, H. (1993). Decisions with Multiple Objectives: Preferences and Value Tradeoffs. Cambridge University Press.

Konno, H. (1990). Piecewise linear risk function and portfolio optimization. Journal of the Operational Research Society of Japan,33(2), 139–156.

Markowitz, H. (1952). Portfolio selection. Journal of Finance,7, 77–91.

Markowitz, H. (1959). Portfolio Selection: Efficient Diversification of Investments. John Wiley &

Sons, New York.

Steuer, R. and Na, P. (2003). Multiple criteria decision making combined with finance: A catego- rized bibliographic study. European Journal of Operational Research,150, 496–515.

Steuer, R., Qi, Y., and Hirschberger, M. (2006). Suitable-portfolio investors, nondominated frontier sensitivity, the effect of multiple objectives on standard portfolio selection.Annals of Operations Research,In print.